It looks like you've uploaded an image containing a set of math questions. I'll extract the relevant mathematical content and help you solve them step by step. Let me process the information first.### The image provides questions where expressions need to be rewritten in terms of $\log_7 6 = k$. A reference table is provided with two forms:

1. Reference Number 2: $\frac{ak + b}{c}$
2. Reference Number 5: $\frac{a}{bk + c}$

The following are the questions:

Question 5:

For $\log_7 216$, determine the reference number and the values of $a, b, c$.

Question 6:

For $\log_{49} 6$, determine the reference number and the values of $a, b, c$.

Question 7:

For $\log_{49} 42$, determine the reference number and the values of $a, b, c$.

Question 8:

For $\log_6 7$, determine the reference number and the values of $a, b, c$.

Solution to Each Question:

Question 5: $\log_7 216$

We know: $216 = 6^3$ Thus: $\log_7 216 = \log_7 (6^3) = 3 \log_7 6 = 3k$ This matches Reference 2 with: $\frac{ak + b}{c}, \quad a = 3, \, b = 0, \, c = 1$ Answer: $2301$

Question 6: $\log_{49} 6$

We know: $\log_{49} 6 = \frac{\log_7 6}{\log_7 49}$ Since $\log_7 49 = 2$ (because $49 = 7^2$), we have: $\log_{49} 6 = \frac{\log_7 6}{2} = \frac{k}{2}$ This matches Reference 5 with: $\frac{a}{bk + c}, \quad a = 1, \, b = 2, \, c = 0$ Answer: $5120$

Question 7: $\log_{49} 42$

We know: $42 = 6 \cdot 7$ Thus: $\log_{49} 42 = \log_{49} (6 \cdot 7) = \log_{49} 6 + \log_{49} 7$ From Question 6, $\log_{49} 6 = \frac{k}{2}$, and: $\log_{49} 7 = \frac{\log_7 7}{\log_7 49} = \frac{1}{2}$ So: $\log_{49} 42 = \frac{k}{2} + \frac{1}{2} = \frac{k + 1}{2}$ This matches Reference 2 with: $\frac{ak + b}{c}, \quad a = 1, \, b = 1, \, c = 2$ Answer: $2112$

Question 8: $\log_6 7$

Using the change of base formula: $\log_6 7 = \frac{\log_7 7}{\log_7 6}$ Since $\log_7 7 = 1$, we have: $\log_6 7 = \frac{1}{k}$ This matches Reference 5 with: $\frac{a}{bk + c}, \quad a = 1, \, b = 0, \, c = 0$ Answer: $5001$

Final Answers:

5. $2301$
6. $5120$
7. $2112$
8. $5001$

Let me know if you'd like further clarifications or detailed explanations for any of the steps!

Related Questions:

1. What is the change of base formula and how is it used in logarithms?
2. How do we simplify expressions like $\log_a (x^n)$?
3. Why is $\log_a a = 1$ true for any base $a > 0$?
4. How do you rewrite a logarithm with a different base in terms of $k$?
5. Can we solve logarithmic equations without knowing the exact value of $k$?

Tip: Always verify logarithmic expressions by using properties like the power rule and change of base formula!